A Numerical Study of the Spectral Gap

We present a numerical study of the spectral gap of the Dirichlet Laplacian, ?(K) = ?2(K) ? ?1(K), of a planar convex region K. Besides providing supporting numerical evidence for the long-standing gap conjecture that ?(K) ? 3?2/d2(K), where d(K) denotes the diameter of K, our study suggests new types of bounds and several conjectures regarding the dependence of the gap not only on the diameter, but also on the perimeter and the area. One of these conjectures is a stronger version of the gap conjecture mentioned above. A similar study is carried out for the quotient of the first two Dirichlet eigenvalues